2.1 The moment maps for parasymplectic spaces

First of all, let $\mathrm G$ be a diffeological group. We denote by ${\mathcal G}^*$ its space of its momenta, that is, the space of the left-invariant differential one-forms on $\mathrm G$ ,

$$ \begin{align*}{\mathcal G}^* = \{\varepsilon \in \Omega^1(\mathrm G) \mid \mathrm{L}(g)^*(\varepsilon) = \varepsilon, \mbox{ for all } g \in \mathrm G \}. \end{align*} $$

Now, let $(\mathrm X,\omega )$ be a parasymplectic space with a parasymplectic action of $\mathrm G$ on $\mathrm X$ . That is, a smooth morphism $\rho :\mathrm G \to \mathrm {Diff}(\mathrm X,\omega )$ , denoted by $g \mapsto g_{\mathrm X}$ , where $\mathrm {Diff}(\mathrm X,\omega ) \subset \mathrm {Diff}(\mathrm X)$ is the group of automorphisms of $\omega $ , equipped with the functional diffeology. Hence, $g_{\mathrm X}^*(\omega )=\omega $ for all $g\in \mathrm G$ .

To understand the essential nature of the moment map, which is a map from $\mathrm X$ to ${\mathcal G}^*$ , it is good to consider the simplest case, and use it then as a guide to extend this simple construction to the general case.

The simplest case. Consider the case where $\mathrm X$ is a manifold, and $\mathrm G$ is a Lie group. Let us assume that $\omega $ is exact, $\omega = d\alpha $ , and that $\alpha $ is also invariant by $\mathrm G$ . Then, the moment map Footnote ³ of the action of $\mathrm G$ on $\mathrm X$ is the map

$$ \begin{align*}\mu : \mathrm X \to {\mathcal G}^* \quad \mbox{ defined by } \quad \mu(x) = \hat x^*(\alpha), \end{align*} $$

where $\hat x : \mathrm G \to \mathrm X$ is the orbit map $\hat x (g)= g_{\mathrm X}(x)$ .

As we can see, there is no obstacle, in this simple situation, to generalize, mutatis mutandis, the Moment Map to a diffeological group acting by symmetries on a diffeological parasymplectic space. However, as we know, not all closed two-forms are exact and, even if they are exact they do not necessarily have an invariant primitive. We shall see now how we can reduce the general case to the simple particular situation by passing to the spaces of paths: $\mathrm {Paths}(\mathrm X) = \mathrm {C}^\infty ({\mathbf R},\mathrm X)$ .

The general case. We consider a connected parasymplectic diffeological space $(\mathrm X,\omega )$ , and a diffeological group $\mathrm G$ acting on $\mathrm X$ and preserving $\omega $ . Let ${\mathcal K}$ be the chain-homotopy operator, defined in [Reference Iglesias-ZemmourPIZ13, Section 6.83]. We recall that

$$ \begin{align*}{\mathcal K} \colon \Omega^k(\mathrm X) \to\Omega^{k-1}(\mathrm{Paths}(\mathrm X)) \end{align*} $$

is a linear operator which satisfies the property

(♡) $$\begin{align} d\circ {\mathcal K} + {\mathcal K} \circ d = \hat 1^* - \hat 0^*, \end{align}$$

where $\hat t(\gamma )=\gamma (t)$ , with $t\in {\mathbf R}$ and $\gamma \in \mathrm {Paths}(\mathrm X)$ . Then, the differential one-form ${\mathcal K}\!\!\omega $ , defined on $\mathrm {Paths}(\mathrm X)$ , is related to $\omega $ by $d[{\mathcal K}\!\!\omega ] = ({\hat 1}^* - {\hat 0}^*)(\omega )$ , and ${\mathcal K}\!\!\omega $ is invariant by $\mathrm G$ (op.cit. Section 6.84). Considering $\bar \omega = ({\hat 1}^* - {\hat 0}^*)(\omega )$ and $\bar \alpha = {\mathcal K}\!\!\omega $ , we are in the simple case: $\bar \omega = d\bar \alpha $ with $\bar \alpha $ invariant. We can apply the construction above and define then the Moment Map of Paths by

$$ \begin{align*}\Psi : \mathrm{Paths}(\mathrm X) \to {\mathcal G}^* \quad \mbox{with} \quad \Psi(\gamma) = \hat \gamma^*({\mathcal K}\!\!\omega), \end{align*} $$

and $\hat \gamma : \mathrm G \to \mathrm {Paths}(\mathrm X)$ is the orbit map $\hat \gamma (g)= g_{\mathrm X} \circ \gamma $ of a path $\gamma $ . The moment of paths is additive with respect to the concatenation [Reference Iglesias-ZemmourPIZ10, Section 4.4],

$$ \begin{align*}\Psi(\gamma \vee \gamma') = \Psi(\gamma) + \Psi(\gamma'). \end{align*} $$

This paths Moment Map $\Psi $ is equivariant by $\mathrm G$ , acting by composition on $\mathrm {Paths}(\mathrm X)$ , and by coadjoint action on ${\mathcal G}^*$ . Next, the Holonomy of the action of $\mathrm G$ on $\mathrm X$ is defined by

$$ \begin{align*}\Gamma = \{\Psi(\ell) \mid \ell \in \mathrm{Loops}(\mathrm X) \} \subset {\mathcal G}^*, \end{align*} $$

the Two-Points Moment Map is defined by pushing $\Psi $ forward on $\mathrm X\times \mathrm X$ ,

$$ \begin{align*}\psi \colon \mathrm X \times \mathrm X \to {\mathcal G}^*\!\!/\Gamma \quad \text{with} \quad \psi(x,x') = \mathrm{class}(\Psi(\gamma)), \end{align*} $$

where $\gamma $ is a path connecting x to $x'$ , and where $\mathrm {class}$ denotes the projection from ${\mathcal G}^*$ onto its quotient ${\mathcal G}^*\!\!/\Gamma $ . The holonomy $\Gamma $ is the obstruction for the action of $\mathrm G$ to be Hamiltonian. The additivity of $\Psi $ becomes the Chasles’ cocycle condition

$$ \begin{align*}\psi(x,x') + \psi(x',x'{'}) = \psi(x,x'{'}). \end{align*} $$

Let $\mathrm {Ad} : \mathrm G \to \mathrm {Diff}(\mathrm G)$ be the adjoint action, $\mathrm {Ad}(g)\colon k \mapsto gkg^{-1}$ . That induces on ${\mathcal G}^*$ a linear coadjoint action

$$ \begin{align*}\mathrm{Ad}_* : \mathrm G \to \mathrm L({\mathcal G}^*) \quad\text{with}\quad \mathrm{Ad}_*(g) : \varepsilon \mapsto \mathrm{Ad}(g)_*(\varepsilon)= \mathrm{Ad}(g^{-1})^*(\varepsilon). \end{align*} $$

Next, the group $\Gamma $ is made of closed forms, invariant by the linear coadjoint action. Thus, the coadjoint action passes to the quotient ${\mathcal G}^*\!\!/\Gamma $ and we denote the quotient action in the same way:

$$ \begin{align*}\mathrm{Ad}_*(g) : \mathrm{class}(\varepsilon) \mapsto \mathrm{class} (\mathrm{Ad}_*(g)(\varepsilon)). \end{align*} $$

The two-points Moment Map $\psi $ is equivariant for the quotient coadjoint action. Note that the quotient ${\mathcal G}^*\!\!/\Gamma $ is in all cases a diffeological Abelian group.

Now, because $\mathrm X$ is connected, there exists always a map

$$ \begin{align*}\mu : \mathrm X \to {\mathcal G}^*\!\!/\Gamma \quad \mbox{such that} \quad \psi(x,x') = \mu(x') - \mu(x). \end{align*} $$

The solutions of this equation are given by

$$ \begin{align*}\mu(x)=\psi(x_0,x) + c, \end{align*} $$

where $x_0 \in \mathrm X$ is an arbitrary point and $c \in {\mathcal G}^*\!\!/\Gamma $ is any constant. This map is a priori no longer equivariant with respect to $\mathrm {Ad}_*$ on ${\mathcal G}^*\!\!/\Gamma $ . Its lack of equivariance defines a one-cocycle $\theta $ of $\mathrm G$ with values in ${\mathcal G}^*\!\!/\Gamma $ :

$$ \begin{align*}\mu(g(x)) = \mathrm{Ad}_*(g)(\mu(x)) + \theta(g), \end{align*} $$

with

$$ \begin{align*}\theta(g) = \psi(x_0,g(x_0)) - \Delta c(g) \quad\text{and}\quad \Delta(c) : g \mapsto \mathrm{Ad}_*(g)(c) - c \end{align*} $$

is the coboundary due to the constant c in the choice of $\mu $ . The cocycle $\theta $ defines then a new action of $\mathrm G$ on ${\mathcal G}^*\!\!/\Gamma $ , that is, a quotient affine action:

$$ \begin{align*}\mathrm{Ad}_*^\theta(g) : \tau \mapsto \mathrm{Ad}_*(g)(\tau) + \theta(g) \quad\text{for all}\quad \tau \in {\mathcal G}^*\!\!/\Gamma. \end{align*} $$

The Moment Map $\mu $ is then equivariant with respect to this affine action:

$$ \begin{align*}\mu(g(x))=\mathrm{Ad}_*^\theta(g)(\mu(x)). \end{align*} $$

Note that, in particular, if $\mathrm G$ is transitive on $\mathrm X$ , then the image of the Moment Map $\mu $ is an affine coadjoint orbit in ${\mathcal G}^*\!\!/\Gamma $ .

This construction extends the Moment Map for {Manifolds} introduced by Souriau in the 60s [Reference SouriauSou70] to the category {Diffeology}.

The group of all automorphisms of a parasymplectic space is naturally a diffeological group, denoted by $\mathrm {Diff}(\mathrm X,\omega )$ or by $\mathrm G_\omega $ . The constructions above give the space of momenta ${\mathcal G}_\omega ^*$ , the universal Footnote ⁴ path moment map $\Psi _\omega $ , the universal holonomy $\Gamma _\omega $ , the universal two-points moment map $\psi _\omega $ , the universal moment maps $\mu _\omega $ , and the universal Souriau’s cocycles $\theta _\omega $ .

2.2 The case of a symplectic manifold

Let $(\mathrm X,\omega )$ be a connected parasymplectic manifold. The value of the paths Moment Map $\Psi _\omega $ at the point $p \in \mathrm {Paths}(\mathrm X) = \mathrm {C}^\infty ({\mathbf R},\mathrm X)$ , evaluated on the n-plot $\mathrm F : \mathrm U \to \mathrm G_\omega $ is explicitely given by [Reference Iglesias-ZemmourPIZ10, Section 10.1]

(◊) $$ \begin{align} \Psi_\omega(p)(\mathrm F)_r(\delta r) = \int_0^1 \omega_{p(t)}(\dot p(t), \delta p(t)) \, dt, \end{align} $$

where $r \in \mathrm U$ and $\delta r \in {\mathbf R}^n$ , $\delta p$ denotes the lifting in the tangent space $\mathrm T\mathrm X$ of the path p, defined by

(♡) $$\begin{align} \delta p(t) = [\mathrm D(\mathrm F(r))(p(t))]^{-1} {\partial \mathrm F(r)(p(t)) \over \partial r} (\delta r) \quad\mbox{for all}\quad t \in {\mathbf R}. \end{align}$$

In that case, we have the following theorem, see [Reference Iglesias-ZemmourPIZ16] for example:

Theorem (P.I-Z) Let $\mathrm X$ be a connected Hausdorff manifold. A parasymplectic form $\omega $ on $\mathrm X$ is symplectic if and only if the following two conditions are satisfied:

1. (1) The manifold $\mathrm X$ is homogeneous under the action of $\mathrm G_\omega $ .

2. (2) The universal Moment Map $\mu _\omega : \mathrm X \to {\mathcal G}^*_\omega /\Gamma _\omega $ is injective.

Hence, the Moment Map identifies the manifold $\mathrm X$ with a, a priori affine, $(\Gamma _\omega ,\theta _\omega )$ -coadjoint orbit ${\mathcal O}_\omega $ of $\mathrm G_\omega $ ,

$$ \begin{align*}\mu_\omega(\mathrm X) ={\mathcal O}_\omega \subset {\mathcal G}^*_\omega/\Gamma_\omega. \end{align*} $$

The situation is summarized by the diagram

(♣)

where $\pi _{\mathrm X} (\phi ) = \phi (x_0)$ , $\pi _{\mathcal O}(\phi ) = \mathrm {Ad}_*^\theta (\phi )(\mu _\omega (x_0))$ , for all $\phi \in \mathrm G_\omega $ and $x_0 \in \mathrm X$ is any base point. The projections $\pi _{\mathrm X}$ is a subduction [Reference BoothbyBoo69, Reference DonatoDon84], ${\mathcal O}_\omega $ is equipped with the pushforward diffeology of $\mathrm G_\omega $ by $\pi _{\mathcal O}$ , and $\mu _\omega $ is then a diffeomorphism.

2.3 Hamiltonian diffeomorphisms

In the construction of the Moment Map of a parasymplectic action of a diffeological group $\mathrm G$ on $(\mathrm X,\omega )$ , the holonomy group $\Gamma $ is the obstruction of the action of $\mathrm G$ to be Hamiltonian.

Hence, the moment maps have their values in ${\mathcal G}^*$ . We get then the following theorem [Reference Iglesias-ZemmourPIZ10, Section 9.2].

The group $\mathrm {Ham}(\mathrm X,\omega )$ is precisely built as follows. Let $\mathrm G_\omega ^\circ $ be the identity component of $\mathrm G_\omega = \mathrm {Diff}(\mathrm X,\omega )$ . Let $\pi : \widetilde {\mathrm G}_\omega ^\circ \to \mathrm G_\omega ^\circ $ be the universal covering. Since the universal holonomy $\Gamma _\omega $ is made of closed momenta [Reference Iglesias-ZemmourPIZ10, Section 4.7], every $\gamma \in \Gamma _\omega $ defines a unique homomorphism $\mathbf {k}(\gamma )$ from $\widetilde {\mathrm G}_\omega ^\circ $ to ${\mathbf R}$ such that $\pi ^*(\gamma ) = d[\mathbf {k}(\gamma )]$ [Reference Iglesias-ZemmourPIZ10, Section 3.11]. Let

$$ \begin{align*}\widehat {\mathbf H}_\omega = \bigcap_{\gamma \in \Gamma_\omega} \ker(\mathbf{k}(\gamma)), \quad\text{then}\quad \mathrm{Ham}(\mathrm X,\omega) = \pi(\widehat {\mathbf H}_\omega^\circ), \end{align*} $$

where $\widehat {\mathbf H}_\omega ^\circ \subset {\mathbf H}_\omega $ is its identity component. The space of momenta and the universal moment maps objects associated to $\mathrm H_\omega = \mathrm {Ham}(\mathrm X,\omega )$ are denoted by: ${\mathcal H}_\omega ^*$ , $\bar \Psi _\omega $ , $\bar \psi _\omega $ , $\bar \mu _\omega $ , and $\bar \theta _\omega $ .

2.4 The integration bundle of a parasymplectic form

Let $(\mathrm X,\omega )$ be a connected, Hausdorff and second countable parasymplectic manifold. Let $\mathrm P_\omega $ its group of periods, that is,

$$ \begin{align*}\mathrm P_\omega = \left\{ \int_\sigma \omega \,\, \bigg| \,\, \sigma \in \mathrm H_2(\mathrm X,{\mathbf Z}) \right\} \subset {\mathbf R}. \end{align*} $$

Since $\mathrm X$ is second countable, $\mathrm P_\omega $ is diffeogically discrete, that is, the diffeology induced by the standard diffeology of ${\mathbf R}$ is the discrete diffeology. The plots are locally constant. Let

$$ \begin{align*}\mathrm T_\omega = {\mathbf R}/\mathrm P_\omega \end{align*} $$

be its torus of periods. Except in the case where the group of periods has only one generator, the torus of periods is not a manifold but nevertheless, a nontrivial diffeological group. See, for example, the paper on the irrational torus $\mathrm T_\alpha = {\mathbf R}/{\mathbf Z} + \alpha {\mathbf Z}$ [Reference Donato and IglesiasDI83], where $\alpha \notin {\mathbf Q}$ .

Then, we get the following theorem in [Reference Iglesias-ZemmourPIZ95, Theorem 1.5].

A connection one-form $\lambda $ on $\mathrm Y$ is a $\mathrm T_\omega $ invariant calibrated one-form, that is, $\tau _{\mathrm Y}^*(\lambda ) = \lambda $ for all $\tau \in \mathrm T_\omega $ and for all $y \in \mathrm Y$ , $\hat y^*(\lambda ) = \theta $ , where $\hat y \colon \mathrm T_\omega \to \mathrm Y$ is the orbit map $\hat y(\tau ) = \tau _{\mathrm Y}(y)$ ; and $\theta $ is the canonical one-form on $\mathrm T_\omega $ defined by $\mathrm {class}^*(\theta ) = dt$ , with $\mathrm {class} \colon {\mathbf R} \to \mathrm T_\omega $ the projection, see [Reference Iglesias-ZemmourPIZ13, Section 8.37].

3.1 The central extension of hamiltonian diffeomorphisms

Let $(\mathrm X,\omega )$ be a symplectic manifold. Let $\mathrm P_\omega $ be its group of periods and $\mathrm T_\omega = {\mathbf R}/\mathrm P_\omega $ be its torus of periods. Let $\pi \colon \mathrm Y \to \mathrm X$ be a $\mathrm T_\omega $ -principal integration fiber bundle, and $\lambda $ be its connection form. Let $\mathrm {Aut}(\mathrm Y,\lambda )$ be the group of automorphisms of $(\mathrm Y,\lambda )$ , that is,

$$ \begin{align*}\mathrm{Aut}(\mathrm Y,\lambda) = \{ \phi \in \mathrm{Diff}(\mathrm Y) \mid \phi^*(\lambda) = \lambda \text{ and } \exists f \in \mathrm{Diff}(\mathrm X),\ \pi \circ \phi = f \circ \pi \}. \end{align*} $$

Actually, we reduce $\mathrm {Aut}(\mathrm Y,\lambda )$ to its identity component.Footnote ⁶ Then, the diffeomorphism f belongs naturally to the group of Hamiltonian diffeomorphisms $\mathrm {Ham}(\mathrm X,\omega )$ . The mapping $\eta \colon \phi \mapsto f$ is then a surjective homomorphism. Its kernel is the torus of periods $\mathrm T_\omega $ , and $\eta $ is a central extension. This is summarized by the exact sequence:

Proof Let us begin by fixing our notation. The action of an element $\tau \in \mathrm T_\omega $ on $y \in \mathrm Y$ will be denoted indifferently by

$$ \begin{align*}\tau \cdot y \quad \text{or by} \quad \tau_{\mathrm Y}(y). \end{align*} $$

Now, let $\phi \in \mathrm {Aut}(\mathrm Y,\lambda )$ and $f = \eta (\phi )$ . Since $f\circ \pi =\pi \circ \phi $ , $\phi ^*(\lambda ) = \lambda $ and $\pi ^*(\omega ) = d\lambda $ , $\pi $ being a subduction, we get $f \in \mathrm {Diff}(\mathrm X,\omega )$ .

(A) Let us to prove that $\ker (\eta )=\mathrm T_\omega $ , acting on $\mathrm Y$ by $\tau \colon y \mapsto \tau \cdot y$ . Let $\phi \in \ker (\eta )$ , that is, $\pi \circ \phi =\pi $ . Then, for all $y\in \mathrm Y$ , there exists a unique $\tau (y) \in \mathrm T_\omega $ such that $\phi (y)=\tau (y) \cdot y$ .

(a) Let us first check that $\tau \colon \mathrm Y \to \mathrm T_\omega $ is smooth. Let $r \mapsto y_r$ by a plot in $\mathrm Y$ , the composite with $\phi $ gives the plot $r \mapsto \tau (y_r) \cdot y_r$ . We need to prove that $r \mapsto \tau (y_r)$ itself is smooth. The pullback of $\pi \colon \mathrm Y \to \mathrm X$ by the plot $r \mapsto x_r = \pi (y_r)$ is locally trivial, then we can restrict these plots to a ball $\mathrm B$ above which the pullback

$$ \begin{align*}[r \mapsto x_r]^*(\mathrm Y)=\{(r,y)\in \mathrm B \times \mathrm Y \mid \pi(y) = x_r \} \end{align*} $$

is trivial. Any $\mathrm T_\omega $ -principal bundle isomorphism $\mathrm F$ from this pullback to the product $\mathrm B \times \mathrm T_\omega $ writes $\mathrm F(r,y) = (r, t(r)(y))$ , and the smooth map t with values in $\mathrm T_\omega $ satisfies the equivariance $t(r)(\tau \cdot y) = \tau \cdot t(r)(y)$ . Thus, $r \mapsto t(r)(y_r)$ is smooth as well as $r \mapsto t(r)(\tau (y_r)(y_r)) = \tau (y_r) \cdot t(r)(y_r)$ . Hence, $r \mapsto \tau (y_r)$ is smooth. Therefore, the function $\tau $ is smooth.

(b) Let us prove now that the function $\tau $ is constant. The invariance $\phi ^*(\lambda )=\lambda $ implies $\lambda (r\mapsto \tau (y_r) \cdot y_r) = \lambda (r\mapsto y_r)$ , for all plots $r \to y_r$ . That is, thanks to the partial derivatives formula [Reference Iglesias-ZemmourPIZ13, Section 8.37 $\clubsuit$ ]

$$ \begin{align*} \lambda(r\mapsto y_r)&=\lambda(r\mapsto\tau(y_r) \cdot y_r)\\ &= \lambda(r\mapsto y_r)+\tau^*(\theta)(r\mapsto y_r), \end{align*} $$

where $\theta $ is the canonical one-form on $\mathrm T_\omega $ . Thus, $\tau ^*(\theta )=0$ . Then $\tau $ is constant. Hence, $\ker \eta =\mathrm T_\omega $ .

(B) Let us prove that $\eta $ takes its values in $\mathrm {Ham}(\mathrm X,\omega )$ . That is, that the holonomy group of $\mathrm {Aut}(\mathrm Y,\lambda )$ vanishes when acting on $(\mathrm X,\omega )$ through the action

$$ \begin{align*}\phi_{\mathrm X}(x) = f(x) \quad \text{with} \quad f = \eta(\phi). \end{align*} $$

We shall denote by ${\mathcal A}^*$ the space of momenta of $\mathrm {Aut}(\mathrm Y,\lambda )$ . The Moment Map $\Psi _{\mathrm X}$ of the action of $\mathrm {Aut}(\mathrm Y,\lambda )$ on $(\mathrm X,\omega )$ is given, according to previous notations by:

$$ \begin{align*}\Psi_{\mathrm X}: \mathrm{Paths}(\mathrm X)\to {\mathcal A}^* \quad \text{with} \quad \Psi(\gamma)=\hat{\gamma}^*({\mathcal K}_{\mathrm X}(\omega)), \end{align*} $$

where $\hat \gamma \colon \phi \mapsto f\circ \gamma $ is the orbit map. Let us prove now that $\Psi _{\mathrm X}(\ell )=\hat {\ell }^* ({\mathcal K}_{\mathrm X}(\omega )) = 0$ , for all $\ell \in \mathrm {Loops}(\mathrm X)$ .

Let us recall, first of all, that the principal fiber bundle $\pi :\mathrm Y\to \mathrm X$ induces, in particular, a subduction of loops spaces:

$$ \begin{align*}\pi_* \colon \mathrm{Loops}(\mathrm Y) \to \mathrm{Loops}(\mathrm X) \quad \text{by pushforward} \quad \pi_*(\underline\ell) = \pi \circ \underline\ell\ , \end{align*} $$

see [Reference Iglesias-ZemmourPIZ13, Section 8.32] and [Reference Iglesias-ZemmourPIZ19]. That is, every plot $r \mapsto \ell _r$ in $\mathrm {Loops}(\mathrm X)$ has a local smooth lifting $r \mapsto \underline \ell _r$ , everywhere, in $\mathrm {Loops}(\mathrm Y)$ . Note that we shall underline the paths in $\mathrm Y$ , to distinguish them from paths in $\mathrm X$ . Now, let $\ell $ and $\underline \ell $ such that $\pi \circ \underline \ell = \ell $ . We have $\hat {\ell }(\phi ) = f \circ \ell = f \circ \pi \circ \underline \ell = \pi \circ \phi \circ \underline \ell =\pi \circ \hat {\underline \ell }(\phi )$ , that is, $\hat \ell = \pi _* \circ \hat {\underline \ell }$ . Thus,

$$ \begin{align*}\hat{\ell}^* ({\mathcal K}_{\mathrm X}(\omega))= (\pi \circ \hat{\underline\ell})^*({\mathcal K}_{\mathrm X}(\omega)) = \hat{\underline\ell}^*((\pi_*)^*({\mathcal K}_{\mathrm X}(\omega))). \end{align*} $$

Then, let us recall the varianceFootnote ⁷ of the chain-homotopy operators ${\mathcal K}_{\mathrm X}$ and ${\mathcal K}_{\mathrm Y}$ , relative to $\mathrm X$ and $\mathrm Y$ [Reference Iglesias-ZemmourPIZ13, Section 6.84], summarized by the commutative diagram:

We have then:

$$ \begin{align*}(\pi_*)^*({\mathcal K}_{\mathrm X}(\omega)) = {\mathcal K}_{\mathrm Y}(\pi^*(\omega)) = {\mathcal K}_{\mathrm Y} (d\lambda). \end{align*} $$

Hence:

$$ \begin{align*}\underline{\hat{\ell}}^*((\pi_*)^*{\mathcal K}_{\mathrm X}(\omega)) = \underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(d\lambda)). \end{align*} $$

Thus:

$$ \begin{align*}\Psi_{\mathrm X}(\ell)=\hat{\ell}^*({\mathcal K}_{\mathrm X}(\omega)) = \underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(d\lambda)) = \Psi_{\mathrm Y}(\underline\ell), \end{align*} $$

where $\Psi _{\mathrm Y}$ is the Moment Map for the action of $\mathrm {Aut}(\mathrm Y,d\lambda )$ on $(\mathrm Y,d\lambda )$ . Note that, that could have been deduced directly from [Reference Iglesias-ZemmourPIZ13, Section 9.13]. Now, according to the fundamental property of the chain-homotopy operator, we have:

$$ \begin{align*} \underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(d\lambda)) + \underline{\hat{\ell}}^*(d({\mathcal K} _{\mathrm Y}(\lambda))) & = \underline{\hat{\ell}}^*(\hat 1^*(\lambda) - \hat 0^*(\lambda))\\ & = (\hat 1\circ\underline{\hat{\ell}})^*(\lambda) - (\hat 0\circ\underline{\hat{\ell}})^*(\lambda) \\ & = 0, \end{align*} $$

because $\underline {\ell }$ is a loop. Therefore,

$$ \begin{align*}\underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(d\lambda)) =-d(\underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(\lambda))). \end{align*} $$

For every plot, $r\mapsto \phi _r$ in $\mathrm {Aut}(\mathrm Y,\lambda )$ , for all r in its domain:

$$ \begin{align*}\underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(\lambda))(\phi_r) = {\mathcal K}_{\mathrm Y}(\lambda)(\phi_r \circ \underline{\ell}) = \int_{{\phi_r}_*(\underline{\ell})}\lambda = \int_{\underline{\ell}}\phi_r^*(\lambda) = \int_{\underline{\ell}} \lambda\,. \end{align*} $$

Hence, $\underline {\hat {\ell }}^* ({\mathcal K}_{\mathrm Y}(\lambda ) )$ is constant, its derivative vanishes and therefore,

(◊) $$ \begin{align} \Psi_{\mathrm X}(\ell) = {\hat\ell}^*({\mathcal K}_{\mathrm X}(\omega)) = \underline{\hat\ell}^*({\mathcal K}_{\mathrm Y}(d\lambda))=0. \end{align} $$

And that complete to prove that $\eta \colon \mathrm {Aut}(\mathrm Y,\lambda ) \to \mathrm {Diff}(\mathrm X,\omega )$ takes its values in $\mathrm {Ham}(\mathrm X,\omega )$ .

(C) Let us show now that $\mathrm T_\omega \subset \mathrm {Aut}(\mathrm Y,\lambda )$ is central, that is, $\eta \colon \mathrm {Aut}(Y,\lambda )\to \mathrm {Ham}(\mathrm X,\omega )$ is a central extension. Let $\phi \in \mathrm {Aut}(\mathrm Y,\lambda )$ . We have seen that $\mathrm T_\omega = \ker (\eta )$ . Thus, for all $\tau \in \mathrm T_\omega $ there exists $\tau ' \in \mathrm T_\omega $ such that $\tau ' = \phi \circ \tau \circ \phi ^{-1}$ . Obviously, $h_\phi \colon \tau \mapsto \tau '$ defines a group isomorphism of $\mathrm T_\omega $ : $h_\phi (\tau _1 \tau _2)=h_\phi (\tau _1)h_\phi (\tau _2)$ , and $h_\phi (\tau )^{-1}=\phi ^{-1}\circ \tau ^{-1}\circ \phi $ .

But $\phi $ is connected to the identity map $\mathbf 1_Y$ via a smooth path $s \mapsto \phi _s \in \mathrm {Aut}(\mathrm Y,\lambda )$ , defined on an open interval ${\mathcal I}$ containing $[0,1]$ , with $\phi _0 = \mathbf 1_Y$ and $\phi _1 = \phi $ . That defines a smooth path of isomorphisms $h_{\phi _s} = \phi _s \circ \tau \circ \phi ^{-1}_s$ . Let us denote $h_s$ for $h_{\phi _s}$ . The map $(s,t) \mapsto h_s(\mathrm {class}(t))$ is a plot defined on ${\mathcal I} \times {\mathbf R}$ , in $\mathrm T_\omega $ . By the monodromy theorem [Reference Iglesias-ZemmourPIZ13, Section 8.25], it has a global lifting $(s,t) \mapsto \mathrm H_s(t)$ , defined on ${\mathcal I} \times {\mathbf R}$ , which is a smooth plot in ${\mathbf R}$ . And the lift is unique with $\mathrm H_0(0) = 0$ .

For every parameter s, the restriction $\mathrm H_s \colon {\mathbf R} \to {\mathbf R}$ is a smooth lifting of the isomorphism $h_s \colon \mathrm T_\omega \to \mathrm T_\omega $ . Thus, up to a constant $b_s \in {\mathbf R}$ , $\mathrm H_s$ is a smooth morphism from ${\mathbf R}$ to ${\mathbf R}$ . Hence, $\mathrm H_s(x) = a_s x + b_s$ , where $s \mapsto a_s$ and $s \mapsto b_s$ are smooth, and $a_s \neq 0$ since $\mathrm H_s$ lifts an isomorphism. Now, for all $s \in {\mathcal I}$ , all $x,x' \in {\mathbf R}$ , $h_s(\mathrm {class}(x+x')) = h_s(\mathrm {class}(x)) + h_s(\mathrm {class}(x'))$ , that is, $\mathrm {class}(\mathrm H_s(x+x')) = \mathrm {class}(\mathrm H_s(x)) + \mathrm {class}(\mathrm H_s(x'))$ , i.e., $a_s(x+x') + b_s = a_sx + b_s + a_sx' + b_s + p$ , and then $b_s \in \mathrm P_\omega $ for all $s \in \mathrm P_\omega $ . Since $s \mapsto b_s$ is smooth and $\mathrm P_\omega $ is (diffeologically) discrete in ${\mathbf R}$ , $b_s$ is constant and equal to $b_0$ which is $0$ . Thus, $\mathrm H_s(x) = a_s x$ . Next, $h_s(\mathrm {class}(x)) = \mathrm {class}(\mathrm H_s(x))$ implies that, for all $p \in \mathrm P_\omega $ there exists $p' \in \mathrm P_\omega $ such that $\mathrm H_s(x + p) = \mathrm H_s(x) + p'$ . That is, $a_s(x+p) = a_s x + p'$ , and then $a_s p \in \mathrm P_\omega $ for all $p \in \mathrm P_\omega $ . Again, since $\mathrm P_\omega $ is discrete in ${\mathbf R}$ , $s \mapsto a_s$ is constant and the lifting $\mathrm H$ writes $\mathrm H_s(x) = ax$ , for some number $a \neq 0$ . Finally, since $H_0$ lifts the identity $h_0 = \mathbf 1_{\mathrm T_\omega }$ by a morphism, $\mathrm H_0(x) = x$ and $a=1$ . Therefore, $\mathrm H_s(x) = x$ for all s and $h_s(\tau ) = \tau $ , that is, $\phi \circ \tau _{\mathrm Y} = \tau _{\mathrm Y} \circ \phi $ , and the extension $\eta \colon \mathrm {Aut}(\mathrm Y,\lambda ) \to \mathrm {Ham}(\mathrm X,\omega )$ is central.

(D) Let us show finally that $\eta :\mathrm {Aut}(\mathrm Y,\lambda ) \to \mathrm {Ham}(\mathrm X,\omega )$ is surjective. Let $f\in \mathrm {Ham}(\mathrm X,\omega )$ . There exists a smooth path $t\mapsto f_t\in \mathrm {Ham}(\mathrm X,\omega )$ , such that $f_0=1_X$ and $f_1=f$ . We define, for all $x\in \mathrm X$ , the path $\gamma _X$ in $\mathrm X$ by $\gamma _x(t)=f_t(x)$ . It satisfies $\gamma _x(0)=x$ and $\gamma _x(1)=f(x)$ . The map $x\mapsto \gamma _x$ is smooth.

Given any $y\in \mathrm Y$ and $x=\pi (y) \in \mathrm X$ , we will denote by $\underline {\gamma }_y$ the unique horizontal lifting of $\gamma _x$ with origin y. Moreover, the map $y\mapsto \underline {\gamma }_y$ is smooth and equivariant under the action of $\mathrm T_\omega $ [Reference Iglesias-ZemmourPIZ13, Section 8.32]. We define then:

$$ \begin{align*}\phi(y)=\underline{\gamma}_y(1) \quad \text{and} \quad \phi \in \mathrm{C}^\infty(\mathrm Y) \quad {({op. cit.})} \end{align*} $$

The map $\phi $ is a smooth lifting of f, that is, $\pi \circ \phi = f \circ \pi $ . Moreover, the equivariance of $\underline {\gamma }_y$ by $\mathrm T_\omega $ also implies that $\tau _{\mathrm Y} \circ \phi = \phi \circ \tau _{\mathrm Y}$ for all $\tau \in \mathrm T_\omega $ . If $\phi $ is equivariant, it has no reason to preserve the contact form $\lambda $ . We shall show then that there exists a map $\tau \in \mathrm {C}^\infty (\mathrm Y, \mathrm T_\omega )$ such that

$$ \begin{align*}\Phi \colon y \mapsto \tau(y) \cdot \phi(y), \end{align*} $$

which is still a smooth lifting of f and preserves the contact form $\lambda $ , that is, $\Phi \in \mathrm {Aut}(\mathrm Y,\lambda )$ . Thanks to the partial derivatives formula (op. cit.), for any plot $r\mapsto y_r$ of $\mathrm Y$ , we get:

$$ \begin{align*} \Phi^*(\lambda)(r\mapsto y_r) & = \lambda(r\mapsto \tau(y_r)\cdot\phi(y_r)) \\ &= \theta(r\mapsto \tau(y_r)) + \lambda(r\mapsto \phi(y_r)) \\ &= \tau^*(\theta)(r\mapsto y_r) + \phi^*(\lambda)(r\mapsto y_r). \end{align*} $$

That is, $\Phi ^*(\lambda )=\tau ^*(\theta )+ \phi ^*(\lambda )$ . Consider now

$$ \begin{align*}\Phi^*(\lambda)-\lambda = \tau^*(\theta) + \beta \quad \text{with} \quad \beta = \phi^*(\lambda) - \lambda. \end{align*} $$

▪

$\blacktriangleleft $ Let us check that $\beta $ is closed: $d(\phi ^*(\lambda )-\lambda ) = \phi ^*(d\lambda )-d\lambda =\phi ^*(\pi ^*\omega )-\pi ^*\omega \ = \pi \circ \phi )^*\omega -\pi ^*\omega =(f\circ \pi )^*\omega -\pi ^* \omega =\pi ^*f^*\omega -\pi ^*\omega =0$ . Also, $\beta $ is invariant by $\mathrm T_\omega $ : $\tau ^*(\beta ) = \tau ^*(\phi ^*(\lambda )-\lambda ) = (\phi \circ \tau )^*(\lambda )-\tau ^*(\lambda ) = (\tau \circ \phi )^*(\lambda )- \lambda = \phi ^*(\tau ^*(\lambda ))-\lambda = \phi ^*(\lambda )-\lambda =\beta $ . Moreover, $\beta $ vanishes vertically. Indeed, let us first remark that $\tau \circ \phi = \phi \circ \tau $ , for all $\tau \in \mathrm T_\omega $ , implies $\phi \circ \hat {y}=\hat {y}'$ , for all $y\in \mathrm Y$ and $y'=\phi (y)$ . Then, $\hat {y}^*(\beta ) = \hat {y}^*(\phi ^*(\lambda )-\lambda ) = \hat {y}^*(\phi ^*(\lambda )-\hat {y}^*(\lambda ) =(\phi \circ \hat {y})^*(\lambda )-\hat {y}^*(\lambda ) = (\hat {y}')^*(\lambda )-\hat {y}^*(\lambda ) = \theta -\theta =0$ . Thus $\lambda ' = \lambda + \beta $ is a new connection one-form, the difference $\beta $ is then the pullback of a one-form on $\mathrm X$ , according to [Reference Iglesias-ZemmourPIZ13, Section 8.37, Note]. $\blacktriangleright $

$\blacktriangleleft $ Indeed, considering the fundamental property of the chain-homotopy operator ${\mathcal K} \circ d + d \circ {\mathcal K} = {\hat 1}^* -{\hat 0}^*$ (Section 2.1 $\heartsuit $ ), on the one hand, and the vanishing of the holonomy of the action of $\mathrm {Aut}(\mathrm Y,\lambda )$ on $\mathrm Y$ (Section 3.1 Proof $\diamondsuit $ ) on the other, we get,

$$ \begin{align*}0 = \Psi_{\mathrm X}(\ell) = \hat\ell^*({\mathcal K}_{\mathrm X}(\omega)) = \hat{\underline\ell}^*({\mathcal K}_{\mathrm Y}(d\lambda)) = - \hat{\underline\ell}^*(d({\mathcal K}_{\mathrm Y}(\lambda))) = -d(\underline{\hat{\ell}}^*({\mathcal K}_{\mathrm Y}(\lambda))), \end{align*} $$

for all $\ell \in \mathrm {Loops}(\mathrm X)$ and all $\underline \ell \in \mathrm {Loops}(\mathrm Y)$ over $\ell $ , because ${\hat 1} \circ \underline \ell ^* = {\hat 0} \circ \underline \ell ^*$ . Now, evaluating the Moment Map on the plot $t\mapsto \phi _t$ connecting $1_Y$ to $\phi $ , using $\ell =\pi \circ \underline {\ell }$ and $\phi _t \circ \underline {\ell } = \phi _{t*}(\underline {\ell })$ , we get:

$$ \begin{align*}\begin{array}{c@{\hspace{5pt}}c@{\hspace{5pt}}c@{\hspace{5pt}}c@{\hspace{5pt}}c@{\hspace{5pt}}c} & d(\hat{\underline\ell}^*({\mathcal K}_{\mathrm Y}(\lambda))(t\mapsto\phi_t)) & = & d({\mathcal K}_Y(\lambda)(t \mapsto \phi_{t*}(\underline{\ell}))) & = & \displaystyle d\left[t \mapsto \int_{\phi_{t*}(\underline{\ell})}\lambda\right]\\[1.2em] = & \displaystyle d\left[t \mapsto \int_{\underline{\ell}}\phi_t^*(\lambda)\right] & = & \displaystyle \int_{\underline{\ell}}\phi^*(\lambda) - \int_{\underline{\ell}}\lambda & = & \displaystyle \int_{\underline{\ell}}\phi^*(\lambda)-\lambda \\[1.2em] = & \displaystyle \int_{\underline{\ell}}\beta & = & \displaystyle \int_{\underline{\ell}}\pi^*\varepsilon & = & \displaystyle \int_\ell \varepsilon. \end{array} \end{align*} $$

Thus, for all $\ell \in \mathrm {Loops}(\mathrm X)$ , $\displaystyle \int _\ell \varepsilon = 0$ . Therefore, according to [Reference Iglesias-ZemmourPIZ13, Section 6.89], there exists $\nu \in C^\infty (X,{\mathbf R})$ such that $\varepsilon = d\nu $ . $\blacktriangleright $

We can now complete to prove that $\Phi \in \mathrm {Aut}(\mathrm Y,\lambda )$ . Indeed, let $\underline \nu = \nu \circ \pi \in \mathrm {C}^\infty (\mathrm Y,{\mathbf R})$ . Let us define $\tau \in C^\infty (\mathrm Y,\mathrm T_\omega )$ by $\tau =-\mathrm {class} \circ \underline \nu = - \mathrm {class} \circ \nu \circ \pi $ , where $\mathrm {class} \colon {\mathbf R} \to \mathrm T_\omega $ . Hence:

$$ \begin{align*}\tau^*(\theta) = -\pi^*(\nu^*(\mathrm{class}^*(\theta))) = \pi^*(\nu^*(dt)) = -\pi^*(d\nu) = - \pi^*(\varepsilon) = -\beta. \end{align*} $$

Thus $\tau ^*(\theta ) = -\phi ^*(\lambda )+\lambda $ . Therefore:

$$ \begin{align*}\Phi^*(\lambda)=\tau^*(\theta)+\phi^*(\lambda)= -\phi^*(\lambda)+\lambda + \phi^*(\lambda) = \lambda{\,}, \quad \text{and}\quad \Phi\in\mathrm{Aut}(\mathrm Y,\lambda). \end{align*} $$

So far, we have proved that $\eta :\mathrm {Aut}(\mathrm Y,\lambda )\to \mathrm {Ham}(\mathrm X,\omega )$ is surjective, we have to prove then that it is a subduction [Reference Iglesias-ZemmourPIZ13, Section 1.46]. For this, we need to check that any plot $P \colon r\mapsto f_r$ in $\mathrm {Ham}(\mathrm X,\omega )$ , admits a local lifting $\tilde {P}$ such that $P =_{\mathrm {locally}} \eta \circ \tilde {P}$ , everywhere. Thanks to the functional diffeology and to both subductions $\pi _* \colon \mathrm {Paths}(Y)\to \mathrm {Paths}(X)$ [Reference Iglesias-ZemmourPIZ13, Section 8.32] and $\pi \colon \mathrm Y \to \mathrm X$ , the map $(r,t,x)\mapsto f_{r,t}(x)$ is smooth and then admits a smooth lifting on Y. Thus, for $x=\pi (y)$ , the time $t=1$ of this lifting defines the smooth family $\phi _r(y)$ of diffeomorphisms, the shift by $\tau \in \mathrm {C}^\infty (Y,\mathrm T_\omega )$ preserves the smoothness of $r\mapsto \Phi _r\in \mathrm {Aut}(\mathrm Y,\lambda )$ .

4.1 Sympectic manifolds as (linear) coadjoint orbits

Let $(\mathrm X,\omega )$ be a symplectic manifold, and as it is described in (Section 3.1), let $\mathrm P_\omega $ be its group of periods, $\pi \colon \mathrm Y \to \mathrm X$ be an integration bundle with connection $\lambda $ , and $\mathrm {Aut}(\mathrm Y,\lambda )$ be the group of automorphisms of the integration structure.

Let ${\mathcal A}^*$ be the space of Momenta of $\mathrm {Aut}(\mathrm Y,\lambda )$ , that is, the space of left-invariant one-forms on $\mathrm {Aut}(\mathrm Y,\lambda )$ . The action of $\mathrm {Aut}(\mathrm Y,\lambda )$ on $\mathrm Y$ has a moment map, relatively to the parasymplectic form $d\lambda $ , given by

$$ \begin{align*}\mu_{\mathrm Y} \colon \mathrm Y \to{\mathcal A}^*, \quad \text{with} \quad \mu_{\mathrm Y}(y) = \hat y^*(\lambda). \end{align*} $$

Then,

1. (1) The moment $\mu _{\mathrm Y}$ descends to $\mu _{\mathrm X}\colon \mathrm X\to {\mathcal A}^*$ , $\mu _{\mathrm Y} = \mu _{\mathrm X}\circ \pi $ .

2. (2) $\mu _{\mathrm Y}$ is equivariant under the coadjoint action of $\mathrm {Aut}(\mathrm Y,\lambda )$ .

3. (3) $\mu _{\mathrm X}$ is injective.

4. (4) $\mu _X$ defines a diffeomorphism from $\mathrm X$ onto the coadjoint orbit

   $$ \begin{align*}{\mathcal A}^* \supset {\mathcal O}_\lambda = \mu_{\mathrm Y}(\mathrm Y)=\mu_{\mathrm X}(\mathrm X). \end{align*} $$

Therefore, the symplectic manifold $\mathrm X$ inherits the structure of a coadjoint orbit. And this is a universal characterization of symplectic manifolds:

This complements the statement made in [Reference Iglesias-ZemmourPIZ16] that Every Symplectic Manifold is a (Affine) Coadjoint Orbit of its group of Symplectomorphisms.

Proof Let us begin by checking that $\mu _{\mathrm Y}$ is constant on each fiber. The action of $\mathrm T_\omega $ is central in $\mathrm {Aut}(\mathrm Y,\lambda )$ , so for any $\tau \in \mathrm T_\omega $ , for all $y\in \mathrm Y$ and for all $\phi \in \mathrm {Aut}(\mathrm Y,\lambda )$ we have: $\widehat {\tau \cdot y}(\phi )= \phi (\tau \cdot y)=\tau \cdot \phi (y)=\tau \cdot (\hat {y}(\phi ))$ , hence $\widehat {\tau \cdot y}=\tau \circ \hat {y}$ . Thus, $\mu _Y(\tau \cdot y) = (\widehat {\tau \cdot y})^*(\lambda ) = (\tau \circ \hat {y})^*(\lambda ) = \hat {y}^*(\tau ^*(\lambda )) = \hat {y}^*(\lambda ) = \mu _Y(y)$ .

Now, let us denote, for all $\phi $ , $\psi $ in $\mathrm {Aut}(\mathrm Y,\lambda )$ , $R(\phi )(\psi )=\psi \circ \phi ^{-1}$ , the right action of the group on its momenta. Then, the equivariance follows from: $\mu _{\mathrm Y}(\phi (y)) = \widehat {\phi (y)}^*(\lambda ) = (\hat {y}\circ \mathrm R(\phi ^{-1})^*(\lambda ) = \mathrm R(\phi ^{-1})^*(\hat {y}^*(\lambda )) = \mathrm R(\phi ^{-1})^*(\mu _{\mathrm Y}(y)) = \mathrm R(\phi ^{-1})^*L(\phi ^{-1})^*(\mu _{\mathrm Y}(y)) = \mathrm {Ad}(\phi ^{-1})^*(\mu _{\mathrm Y}(y)) = \mathrm {Ad}(\phi )_*(\mu _{\mathrm Y}(y))$ .

Finally, pushing forward the moment maps [Reference Iglesias-ZemmourPIZ13, Sections 9.12 and 9.13] leads to the commutative diagram below, where $\bar \mu _{\mathrm X}$ is the Moment Map for the group $\mathrm {Ham}(\mathrm X,\omega )$ , and ${\mathcal H}^*$ denotes its space of momenta.

The Moment Map $\bar \mu _{\mathrm X}$ is known to be injective [Reference Iglesias-ZemmourPIZ16], as well as $\eta ^*$ since $\eta $ is a subduction (Section 3.1, Proof (C)). Hence, $\mu _{\mathrm X}=\eta ^*\circ \bar \mu _{\mathrm X}$ is injective and a subduction on ${\mathcal O}_\lambda = \mu _{\mathrm Y}(\mathrm Y)$ . Therefore, $\mu _{\mathrm X}$ is a diffeomorphism from $\mathrm X$ to ${\mathcal O}_\lambda = \mu _{\mathrm Y}(\mathrm Y)$ , equipped with the quotient diffeology of $\mathrm {Aut}(\mathrm Y,\lambda )$ by the stabilizer of any point $y \in \mathrm Y$ .▪

4.2 The case of a lie group action

For the reader who missed that point, let us show first how the moment map in diffeology [Reference Iglesias-ZemmourPIZ10] generalizes the definition given originally by Souriau [Reference SouriauSou70]. Let $(\mathrm X,\omega )$ be a symplectic manifold and let $\mathrm G$ be a connected Lie group acting on $\mathrm X$ and preserving $\omega $ . In that case, the space of momenta ${\mathcal G}^*$ is the dual of the Lie algebra ${\mathcal G}$ , represented by the space of invariant vector fields, or equivalently, by the set of one-parameter subgroups. Let $h \colon s \mapsto \exp (s\mathrm Z)$ be a one-parameter subgroups, $\mathrm Z$ belonging to the tangent space at the identity identified with ${\mathcal G}$ . Coming back to the expression of the path moment map $\Psi $ , the variation (Section 2.2 $\heartsuit $ ) becomes, for $\delta s = 1$ ,

$$ \begin{align*} \delta p(t) & = [\mathrm D(\exp(s\mathrm Z)))(p(t))]^{-1} {\partial \exp(s\mathrm Z)(p(t)) \over \partial s} \\ & = \left.{\partial \exp(s\mathrm Z)(p(t)) \over \partial s}\right|{}_{s=0} \\ & = \mathrm Z_{\mathrm X}(p(t)), \end{align*} $$

where $\mathrm Z_{\mathrm X}$ is the infinitesimal action of h on $\mathrm X$ . Thus, the expression of the path-moment map (Section 2.2 $\diamondsuit $ ) writes

$$ \begin{align*}\Psi_\omega(p)(h)_s(1) = \int_0^1 \omega_{p(t)}(\dot p(t), \mathrm Z_{\mathrm X}(p(t))) \, dt. \end{align*} $$

The classical moment map $\mu $ is defined in [Reference SouriauSou70, Section 11.7], as a solution, at least locally, of the differential equation

$$ \begin{align*} \omega(\mathrm Z_{\mathrm X}(x)) = - d(\mu(x) \cdot \mathrm Z). \end{align*} $$

We assume now that $\mu $ is defined globally, that is, the action of $\mathrm G$ is Hamiltonian. Thus,

$$ \begin{align*}\Psi_\omega(p)(h)_s(1) = \int_0^1 \left.{\partial \mu(x)\cdot \mathrm Z \over \partial x}\right|_{x=p(t)}\left({d p(t) \over dt}\right) \, dt = (\mu(x') - \mu(x))\cdot \mathrm Z. \end{align*} $$

Hence, $\Psi (p) = \psi (x,x') = \mu (x') - \mu (x)$ . Therefore, since this equation has a unique solution, up to a constant, the moment map in diffeology coincides with the classical moment map when $\mathrm X$ is a manifold and $\mathrm G$ is a Lie group.

Now, let us assume that the action of $\mathrm G$ on $\mathrm X$ is transitive. Souriau proved in [Reference SouriauSou70, Section 11.38] that the moment map is a covering onto its image. As it is explicitly shown in the example of the “Cylinder and $\mathop {\mathrm {SL}}\nolimits (2,{\mathbf R})$ ” [Reference Iglesias-ZemmourPIZ16, Section 7], this covering can be nontrivial. The group $\mathop {\mathrm {SL}}\nolimits (2,{\mathbf R})$ acts transitively on the cylinder ${\mathbf R}^2 - \{0\}$ preserving the symplectic form $\mathrm {Surf} = dx \wedge dy$ . And the moment map is given by

$$ \begin{align*}\mu(z)(\mathrm F_\sigma) = \frac{1}{2} \mathrm{Surf}(z,\sigma z) \times dt, \end{align*} $$

where $z = (x,y) \in {\mathbf R}^2 - \{0\}$ , $\mathrm F_\sigma = [s \mapsto \exp (s\sigma )]$ is the one-parameter group defined by $\sigma \in \mathop {\mathfrak {sl}}\nolimits (2,{\mathbf R})$ , the Lie algebra of $\mathop {\mathrm {SL}}\nolimits (2,{\mathbf R})$ , vector space of real $2 \times 2$ traceless matrices. We have clearly $\mu (z) = \mu (-z)$ .

So, why is there a discrepancy between the Hamiltonian Lie group situation, where the moment map is a covering but may not be injective, and the full Hamiltonian group, for which the moment map is injective? As we can see in [Reference Iglesias-ZemmourPIZ16, Section 3 Proof A’], a key ingredient for the injectivity of the universal moment map is the existence of compactly supported functions that separate points. These are the Hamiltonian functions of the one-parameter groups generated by their gradient, against which the moment map is tested. In this example, in particular, the Hamiltonian functions of the one-parameter subgroups of $\mathop {\mathrm {SL}}\nolimits (2{\mathbf R})$ are exactly the functions $f_\sigma \colon z \mapsto \mu (z)(\mathrm F_\sigma )$ , for all $\sigma \in \mathop {\mathfrak {sl}}\nolimits (2,{\mathbf R})$ , and they do not separate opposite points. So, with a Lie group we may not have enough hamiltonian functions to separate the points of the symplectic manifold; this does not happen with the whole group of Hamiltonian diffeomorphisms.